Problem 18 - Algebra for College Students

Question

Solve each equation. \(9 y+3=4 y-10\)

Step-by-Step Solution

Answer

\(y = -\frac{13}{5}\)

Step 1: Move Variables to One Side

First, subtract \(4y\) from both sides of the equation to get all the \(y\) terms on one side. This gives us:\[9y - 4y + 3 = -10\]This simplifies to:\[5y + 3 = -10\]

Step 2: Isolate the Variable Term

Next, subtract \(3\) from both sides of the equation to isolate the \(5y\) term. This yields:\[5y = -10 - 3\]This simplifies to:\[5y = -13\]

Step 3: Solve for y

Now, divide both sides by \(5\) to solve for \(y\):\[y = \frac{-13}{5}\]

Step 4: Simplify the Fraction

The fraction \(\frac{-13}{5}\) is already in its simplest form, so the solution remains:\[y = -\frac{13}{5}\]

Key Concepts

Equation Simplification

When solving linear equations, our first goal is to simplify the equation. Simplification involves combining like terms and making the equation more straightforward.
In the exercise, this process began by moving variables to one side of the equation. Subtracting the same variable term from both sides keeps the equation balanced, a key principle in algebra.
For example, in the equation given, we started with:

• Original Equation: \[9y + 3 = 4y - 10\]

Subtracting \(4y\) from both sides helps simplify the number of terms:

• Simplified Equation: \[5y + 3 = -10\]

This step reduces the complexity, making the equation easier to solve in the next stages of the process.

Variable Isolation

Variable isolation refers to the process of getting the variable by itself on one side of the equation. This is crucial for finding its value.
In our problem, after simplifying the equation, we focus on the term with the variable \(y\). Here's what it looked like after simplification:

• Simplified Equation: \[5y + 3 = -10\]

We need to isolate the \(5y\) term. To achieve this, subtract \(3\) from both sides of the equation:

• Equation after subtracting 3: \[5y = -13\]

At this point, the term \(5y\) stands alone, and you can easily identify the next step - solving for \(y\). We realize how each operation keeps the equation in balance while focusing on isolating the variable.

Fraction Simplification

Fraction simplification is the final step when you end up with a fractional expression after isolating the variable. It's all about making sure the fraction is in its simplest form.
After isolating \(y\), our solution came down to solving:

• Initial Fraction Equation: \[y = \frac{-13}{5}\]

To determine if the fraction needs simplification, check if the numerator and the denominator have any common factors.
Since \(-13\) and \(5\) share no common factors other than 1, the fraction is already simplified. Thus, it remains as:

• Simplified Fraction: \[y = -\frac{13}{5}\]

This step confirms our solution is fully minimized, ensuring accuracy in your answer.